I want to know a good estimate for the upper bound of the following, $$\nu(x+y)-\nu(x)$$ where $\nu(x)=\sum_{p\leq x}\log p$. It would be of great help if anyone could give me some references.
Thanks.
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1$\begingroup$ For upper bounds, you are looking for the Brun Titchmarsh Theorem. For lower bounds and asymptotics, see prime gaps: en.wikipedia.org/wiki/Prime_gap#Upper_bounds $\endgroup$– Eric NaslundCommented Nov 29, 2012 at 15:53
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$\begingroup$ For clarity, you might add that the sum is over primes p less than or equal to x. Gerhard "Ask Me About System Design" Paseman, 2012.11.29 $\endgroup$– Gerhard PasemanCommented Nov 29, 2012 at 16:19
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$\begingroup$ Also posted to m.se, math.stackexchange.com/questions/247289/… $\endgroup$– Gerry MyersonCommented Nov 30, 2012 at 1:52
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Adding to Eric Naslund's comment: Let $\pi(x)$ denote the number of primes less than $x$, then Montgomery & Vaughan proved that $$ \pi(x+y)-\pi(x) \le 2 \pi(y)$$ for $x\ge 1$ and $y\ge 2.$
answered Nov 29, 2012 at 15:59
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1$\begingroup$ @Micah: I believe you need to have $y\geq 2$ so that the right hand side is nonzero. $\endgroup$ Commented Nov 29, 2012 at 16:04